Reentrant ferromagnetic ordering of the random-field Heisenberg model in $d>2$ dimensions: Fourier-Legendre renormalization-group theory

The random-magnetic-field classical Heisenberg model is solved in spatial dimensions $d\ge 2$ using the recently developed Fourier-Legendre renormalization-group theory for $4\pi$ steradians continuously orientable spins, with renormalization-group flows of 12 500 variables. The random-magnetic-field Heisenberg model is exactly solved in 10 hierarchical models, for $d=2,2.26,2.46,2.58,2.63,2.77,2.89,3$. For nonzero random fields, ferromagnetic order is seen for $d>2$. This ordering, at $d=2.46,2.58,2.63,2.77,2.89,3$, shows reentrance as a function of temperature.